Course syllabus for Differential calculus

• Explain basic concepts in set theory and real analysis, such as real numbers, sequences of real numbers, convergence, and simple properties of subsets of the real numbers, and use these to analyse simple mathematical problems, as well as give an overview of how they relate to the computer representation of real numbers.

• Understand and use the concept of a function and analyse key properties of functions of one variable, in particular elementary functions, using algebraic and graphical methods.

• Formulate and apply the concepts of limit and continuity for functions of one real variable, and determine and interpret limits using analytical and numerical methods.

• Formulate and apply the concepts of derivative and differentiability, together with basic rules of differentiation, for functions of one real variable, and use derivatives to analyse local and global properties of functions and for linearisation.

• Use Taylor polynomials to approximate functions of one real variable, and determine limits.

• Formulate scalar equations as mathematical models for simple engineering problems and solve them using basic numerical methods, and perform a basic analysis of errors and convergence.

• Implement and use numerical algorithms in Python for computations within the scope of the course, and analyse, interpret, and communicate in writing modelling choices, computational results, and limitations in engineering applications.

Real numbers
Set theory and logic; rational and real numbers; sequences and Cauchy sequences; supremum/infimum; open, closed, and bounded sets; computer representation of real numbers (IEEE 754, machine precision).

Functions
The concept of a function; injectivity/surjectivity/bijectivity; inverse and restriction; algebra of functions and composition; polynomials and rational functions; elementary functions (power, exponential, and logarithmic functions, trigonometric functions and their inverses).

Limits and continuity
Definition of limit and continuity; uniform and Lipschitz continuity; symbolic determination of limits (standard limits, rewriting and algebraic manipulation); numerical determination (Richardson extrapolation).

Derivative and linearisation

Definition of the derivative; derivatives of elementary functions; rules of differentiation (sum, product, quotient, chain rule, inverse functions); extrema; the Mean Value Theorem; linearisation and error estimates; numerical differentiation and choice of step size; application of derivatives to determine where a function is increasing/decreasing and convex/concave; critical points and local extreme points; optimisation; implicit differentiation.

Taylor polynomials and series
Taylor and Maclaurin polynomials; determination of limits using big-O notation and L’Hôpital’s rule; Taylor series.

Numerical solution of equations
Equations, roots, and fixed points; the bisection algorithm and the Intermediate Value Theorem; fixed-point iteration and the Banach fixed point theorem; Newton’s method; order of convergence and conditions for convergence.

Applications and programming
Basic building blocks of a Python program. Python as a computational tool; implementation, testing, and validation; for example, computation of elementary functions and simulation of simple mechanical systems.